Server GPU's DO exist. (Well, not in a way you think.) A "Server GPU" is a name or label given to Compute cards designed to do multiple complicated calculations but don't actually render them (some systems have a the render unit together with the compute unit though). Therefore, it is not optimized for gaming. These cards are designed to do heavy computation and processing that are sent to a render unit to render and display the information. If you put these cards into a gaming computer then the cards will have to do something it isn't designed to do which is render an output. Supercomputers usually have multiple computers clustered together with these cards that calculate BIG numbers. And as you expect not all of these computers will have a display attached to them. The data these cards process are sent somewhere else to be rendered and displayed. And to be clear, server "GPU's" are not GPU's at all since they do not process graphics but they process data. The proper name for them is "Compute Cards"
A short answer to your question is, no, the GTX 580 is NOT a server card. It's a graphics card meant for gaming (in 2010, at least). You should get a 960 or 1050 Ti instead. Only a few go used so you could find maybe a 750 Ti used for cheap.